I have a collection of scars from periods of cutting while drinking, sometimes cutting because I was ashamed of the drinking. Even once ended up in the loony bin for 3 days following a long period of drinking and a cutting episode a few years ago. And you think that would have been my bottom, rather than a weepy episode after a pool party.
Cutting, from my experience, is about release and about shame. We feel shame, so we cut to punish ourselves. Maybe he is upset about the breakup, and maybe ashamed at his own behavior in the relationship that could have led to problems. I benefited from a course of antidepressants and therapy, and I can absolutely tell you that drinking does not make the shame go away, it doesn't make anything better, and it makes it much, much worse.